A1 Journal article (refereed)
Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance (2020)

Franks, J., & Vihola, M. (2020). Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance. Stochastic Processes and Their Applications, 130(10), 6157-6183. https://doi.org/10.1016/j.spa.2020.05.006

JYU authors or editors

Publication details

All authors or editors: Franks, Jordan; Vihola, Matti

Journal or series: Stochastic Processes and Their Applications

ISSN: 0304-4149

eISSN: 1879-209X

Publication year: 2020

Volume: 130

Issue number: 10

Pages range: 6157-6183

Publisher: Elsevier

Publication country: Netherlands

Publication language: English

DOI: https://doi.org/10.1016/j.spa.2020.05.006

Publication open access: Not open

Publication channel open access: 

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/69233

Publication is parallel published: https://arxiv.org/abs/1706.09873

Abstract

We establish an ordering criterion for the asymptotic variances of two consistent Markov chain Monte Carlo (MCMC) estimators: an importance sampling (IS) estimator, based on an approximate reversible chain and subsequent IS weighting, and a standard MCMC estimator, based on an exact reversible chain. Essentially, we relax the criterion of the Peskun type covariance ordering by considering two different invariant probabilities, and obtain, in place of a strict ordering of asymptotic variances, a bound of the asymptotic variance of IS by that of the direct MCMC. Simple examples show that IS can have arbitrarily better or worse asymptotic variance than Metropolis–Hastings and delayed-acceptance (DA) MCMC. Our ordering implies that IS is guaranteed to be competitive up to a factor depending on the supremum of the (marginal) IS weight. We elaborate upon the criterion in case of unbiased estimators as part of an auxiliary variable framework. We show how the criterion implies asymptotic variance guarantees for IS in terms of pseudo-marginal (PM) and DA corrections, essentially if the ratio of exact and approximate likelihoods is bounded. We also show that convergence of the IS chain can be less affected by unbounded high-variance unbiased estimators than PM and DA chains.

Keywords: stochastic processes; Markov chains; Monte Carlo methods; estimating (statistical methods); numerical methods

Free keywords: asymptotic variance; delayed-acceptance; importance sampling; Markov chain Monte Carlo; pseudo-marginal algorithm; unbiased estimator

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Ministry reporting: Yes

Reporting Year: 2020

JUFO rating: 2